Integrating Factor for First Order ODE/Examples

Examples of Integrating Factor for First Order ODE

This page gathers examples of first order ODEs solved using the technique of Solution of First Order ODE by Integrating Factor.

$y \rd x + \paren {x^2 y - x} \rd y = 0$

The first order ODE:

$(1): \quad y \rd x + \paren {x^2 y - x} \rd y = 0$

has the general solution:

$\dfrac {y^2} 2 - \dfrac y x = C$

$\paren {3 x^2 - y^2} \rd y - 2 x y \rd x = 0$

The first order ODE:

$(1): \quad \paren {3 x^2 - y^2} \rd y - 2 x y \rd x = 0$

has the general solution:

$\dfrac 1 y - \dfrac {x^2} {y^3} = C$

$\paren {x y - 1} \rd x + \paren {x^2 - x y} \rd y = 0$

The first order ODE:

$(1): \quad \paren {x y - 1} \rd x + \paren {x^2 - x y} \rd y = 0$

has the general solution:

$x y - \ln x - \dfrac {y^2} 2 + C$

$y \rd x + x \rd y + 3 x^3 y^4 \rd y = 0$

The first order ODE:

$(1): \quad y \rd x + x \rd y + 3 x^3 y^4 \rd y = 0$

has the general solution:

$-\dfrac 1 {2 x^2 y^2} + \dfrac {3 y^2} 2 = C$